Momentum space function

The transformations between coordinate and momentum space are accomplished via the fast Fourier transform (FFT) algorithm. This transforms the coordinate space evaluation of exp( — iFAt/2ft) into momentum space the effect of exp( — iTAt/h) is determined in this space to yield the momentum space function the momentum space function is transformed back to a coordinate space function via another FFT and finally, the new coordinate space function is further evolved by the local operator exp( — jFAt/21i) to yield the coordinate space wavefunction at time t-F At. The analogous technique can be accomplished with the simpler formula of Eq. (4.6a), but this does not reduce the number of FFT evaluations per time step, and thus is of no utility. This is clear when one notes that the values of < exp( —t FAt/2/t) > and

are evaluated at the grid points (in the FFT) just once at the beginning of the calculation and stored. Essentially all the work is involved in the FFT computation. 

As we have already mentioned above, for 5-wave scattering the function/(r,R) depends only on the absolute values of r and R and on the angle between them. Thus, Equation 10.24 is an integral equation for the function of three variables. In order to find the molecule-molecule scattering length, it is more convenient to transform Equation 10.24 into an equation for the momentum-space function, /(k,p) = / d rd R/(r,R)exp(ik r/a-l-ip R/ s/2a), which yields the following expression ... 

What has been sketched here is obviously just the bare framework of a general investigation of the symmetry properties of momentum space functions in quasicrystals. With all the information available in the papers by Mermin and collaborators it should however be a very tempting enterprise to go ahead along the lines sketched and learn about the details of the symmetry properties of those wave functions - both in momentum and in positition space - which will be needed in quasiperiodic extended systems. 

Let s first look at the momentum space part. For the momentum space function Hn we have... 

The transformations connecting the coordinate-space wave function, v[/(R), to the momentum-space wave function, k), are... 

The discretized momentum-space wave function corresponding to a momentum of ki% is denoted by 44. As with the discretized spatial wave function [Eq. (37)], the discretized momentum wave functions are also normalized so that 4/ p = 1 (i.e., = i/ ki) V ). 

The wave function in momentum space is given by the Fourier transform of the coordinate-space wave function... 

The discretized momentum space wave function, is therefore given by... 

Mermin s "generalised crystallography" works primarily with reciprocal space notions centered around the density and its Fourier transform. Behind the density there is however a wave function which can be represented in position or momentum space. The wave functions needed for quasicrystals of different kinds have symmetry properties - so far to a large extent unknown. Mermin s reformulation of crystallography makes it attractive to attempt to characterise the symmetry of wave functions for such systems primarily in momentum space. 

In the next setion we review some key concepts in Mermin s approach. After that we summarise in section III some aspects of the theory of (ordinary) crystals, which would seem to lead on to corresponding results for quasicrystals. A very preliminary sketch of a study of the symmetry properties of momentum space wave functions for quasicrystals is then presented in section IV. 

The symmetry properties of the momentum space wave functions can be obtained either from their position space counterparts or more directly from the counterpart of the Hamiltonian in momentum space. 

An important and interesting question is obviously whether for quasicrystals and incommensurately modulated crystals there is anything corresponding to the Bloch functions for crystals. Momentum space may be a better hunting ground in that connection than ordinary space, where we have no lattice. Not only is there no lattice, one cannot even specify the location of each atom yet [8]. 

A Bloch function for a crystal has two characteristics. It is labeled by a wave vector k in the first Brillouin zone, and it can be written as a product of a plane wave with that particular wave vector and a function with the "little" period of the direct lattice. Its counterpart in momentum space vanishes except when the argument p equals k plus a reciprocal lattice vector. For quasicrystals and incommensurately modulated crystals the reciprocal lattice is in a certain sense replaced by the D-dimensional lattice L spanned by the vectors It is conceivable that what corresponds to Bloch functions in momentum space will be non vanishing only when the momentum p equals k plus a vector of the lattice L. 

E. Weigold, Momentum Space Wave Functions. American Institute of Physics, vol. 86, Adelaide, 1982... 

The transform A(p, t) is ealled the momentum-space wave function, while (jc, /) is more accurately known as the coordinate-space wave function. When there is no confusion, however, (jc, /) is usually simply referred to as the wave function. 

The expectation value p) of the momentum p may be obtained using the momentum-space wave function A p, i) in the same way that (x) was obtained from F(x, i). The appropriate expression is... 

What is the probability density as a function of the momentum p of an oscillating particle in its ground state in a parabolic potential well (First find the momentum-space wave function.)... 

Show that the wave functions A (y) in momentum space corresponding to 0 ( ) in equation (4.40) for a linear harmonic oscillator are... 

Using the valence profiles of the 10 measured directions per sample it is now possible to reconstruct as a first step the Ml three-dimensional momentum space density. According to the Fourier Bessel method [8] one starts with the calculation of the Fourier transform of the Compton profiles which is the reciprocal form factor B(z) in the direction of the scattering vector q. The Ml B(r) function is then expanded in terms of cubic lattice harmonics up to the 12th order, which is to take into account the first 6 terms in the series expansion. These expansion coefficients can be determined by a least square fit to the 10 experimental B(z) curves. Then the inverse Fourier transform of the expanded B(r) function corresponds to a series expansion of the momentum density, whose coefficients can be calculated from the coefficients of the B(r) expansion. 

The function 4> k) is known as the wave function in momentum space. The Fourier integral represents the superposition of many waves of different wave vectors. This construct defines a wave packet, once considered as the theoretically most acceptable description of a wave-mechanical particle5. Schrodinger s dynamical equation (4) for a free particle... 

Figure 5.1 illustrates the Fourier transform (FT) of a simple function, viz., a Gaussian. The relatively sharp Gaussian function with the exponent a = 1 depicted in Figure 5.1a, yields a diffuse Gaussian (in dotted line) in momentum space. A flat Gaussian function in position space with a = 0.1, transforms to a sharp one (cf. Figure 5.1b). Connected by an FT, the wave functions in position and momentum...

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